Grothendieck-Lidskitheorem for subspaces of

quotients of Lp-spaces Oleg Reinov and Qaisar Latif

Abstract. Generalizing A. Grothendieck's (1955) and V.B. Lidski's(1959) trace formulas, we have shown in a recent paper that for p ∈ [1, ∞] and s ∈ (0, 1] with 1/s = 1 + |1/2 − 1/p| and for every s- T in every subspace of any Lp(ν)-space the trace of T is well dened and equals the sum of all eigenvalues of T. Now, we obtain the analogues results for subspaces of quotients (equivalently: for quotients of subspaces) of Lp-spaces.

In the note [13], we have proved that if p ∈ [1, ∞] and 1/s = 1+|1/2−1/p|, then for any subspace (or quotient) of an Lp-space and for every s-nuclear operator T in the space the nuclear trace of T is well-dened and equals the sum of all eigenvalues of T. The main fact, we are going to obtain here, is Theorem. Let Y be a subspace of a quotient (or a quotient of a subspace) of an Lp-space, 1 ≤ p ≤ ∞. If T ∈ Ns(Y,Y )(s-nuclear), where 1/s = 1 + |1/2 − 1/p|, then 1. the∑ (nuclear) trace of T is well dened, 2. ∞ | | ∞ where { } is the system of all eigenvalues of the n=1 λn(T ) < , λn(T ) operator T (written in according to their algebraic multiplicities) and ∑∞ trace T = λn(T ). n=1 Let us mention that in the proof we have to repeat some ideas of proofs from [13] (in particular, of the proof of main lemma there) as well as, simultaneously, to use the main lemma [13] itself (so, we will get a generalization of the lemma by using a part of its proof and also its statement). Ÿ1. Preliminaries. All the terminology and facts we use here can be found in [58]. ∗ b Let X,Y be Banach spaces. For s ∈ (0, 1], denote by X ⊗sY the completion of the tensor product X∗ ⊗ Y (considered as a linear space of all nite rank operators)

The research was supported by the Higher Education Commission of Pakistan. AMS Subject Classication 2010: 47B06. Key words: s-nuclear operators, eigenvalue distributions. 1  GROTHENDIECK-LIDSKII THEOREM FOR SUBSPACES OF QUOTIENTS OF Lp-SPACES 2 with respect to the quasi- ( ) ∑N 1/s ∑N || || { || ′ ||s || ||s ′ ⊗ } z s := inf xk yk : z = xk yk . k=1 k=1

Let Ψp, for p ∈ [1, ∞], be the ideal of all operators which can be factored through ∗ b a subspace of a quotient of an Lp-space. Put Ns(X,Y ) := image of X ⊗sY in the space L(X,Y ) of all bounded linear transformations under the canonical factor map ∗ b X ⊗sY → Ns(X,Y ) ⊂ L(X,Y ). We consider the (Grothendieck) space Ns(X,Y ) of all s-nuclear operators from X to Y with the natural quasi-norm, induced from ∗ b X ⊗sY. Finally, let Ψp,s be the quasi-normed product Ns ◦Ψp of the corresponding ideals equipped with the natural quasi-norm νp,s: if A ∈ Ns ◦ Ψp(X,Y ) then A = φ ◦ T with T = βα ∈ Ψp, φ = δ∆γ ∈ Ns and

α β γ ∆ δ A : X → Xp → Z → c0 → l1 → Y, where all maps are continuous and linear, Xp is a subspace of a quotient of an Lp-space, constructed on a space, and ∆ is a diagonal operator with the diagonal from ls. Thus, A = δ∆γβα and A ∈ Ns. Therefore, if X = Y, the spectrum of A, sp (A), is at most countable with only possible limit point zero. Moreover, A is a Riesz operator with eigenvalues of nite algebraic multiplicities and sp (A) ≡ sp (B), where B := αδ∆γβ : Xp → Xp is an s-nuclear operator, acting in a subspace of a quotient of an Lp-space. Denition. Let Y be a and s ∈ (0, 1]. We say that Y possesses the property APs ( the of order s; written down as "Y ∈ APs") ∗ b if for any tensor element z ∈ Y ⊗sY the operator z˜ : Y → Y, associated with z, is zero i the tensor element z is zero as an element of the space Y ∗⊗bY. ∗ b This is equivalent to the fact that if z ∈ Y ⊗sY then it follows from

trace z ◦ R = 0, ∀ R ∈ Y ∗ ⊗ Y that trace U ◦ z = 0 for every U ∈ L(Y,Y ∗∗). There is a simple characterization of the condition Y ∈ APs in terms of the approximation of the identity idY on some of the space Y, but we omit it. We need here only some examples which are crucial for our note. For giving them, we formulate and prove the following statement, which, we hope, is interesting by itself. Proposition 1. Let α ∈ [0, 1/2] and 1/s = 1 + α. For a Banach space Y, suppose that (α) there exist constants C > 0 such that for every ε > 0, for every natural n and for every n-dimensional subspace E of Y there exists a nite rank operator R α in Y so that ||R|| ≤ Cn and ||R|E − idE ||L(E,Y ) ≤ ε. Then Y ∈ APs.  GROTHENDIECK-LIDSKII THEOREM FOR SUBSPACES OF QUOTIENTS OF Lp-SPACES 3 ∗ b Proof. Suppose that there is an element z ∈ Y ⊗sY such that trace z = b > 0, but z˜ = 0. Consider a representation of z of the kind ∑∞ ′ ⊗ z = µkyk yk, k=1 ∑ where || ′ || || || and ≥ ∞ s ∞ Without loss of generality, we can yk , yk = 1 µk 0, k=1 µk < . ∑ (and do) assume that the sequence is decreasing and that ∞ ≤ In this (µk) k=1 µk 1. situation, s so, there are with → and ≤ 1/s µk = o(1/k), ck > 0 ck 0 µk ck/k . Fix any natural N, large enough, such that for all m ≥ N ∑m ⟨ ′ ⟩ ≥ µk yk, yk b/2. k=1 For such an put { }m and apply the condition to nd a corre- m, E := span yk k=1, (α) sponding operator R ∈ Y ∗ ⊗ Y for n = m and ε = b/4. By our assumption, trace R ◦ z = 0. From this, we get (for all m ≥ N): ∑m ∑∞ ⟨ ′ ⟩ ⟨ ′ ⟩ 0 = µk yk, Ryk + µk yk, Ryk . k=1 k=m+1 For the rst sum: ∑m ∑m ∑m ⟨ ′ ⟩ ≥ ⟨ ′ ⟩ − ⟨ ′ − ⟩ ≥ − µk yk, Ryk µk yk, yk µk yk, yk Ryk b/2 b/4 = b/4. k=1 k=1 k=1 For the second sum: ∫ ∑∞ ∞ ⟨ ′ ⟩ ≤ α −1/s ≤ α 1−1/s µk yk, Ryk Cm c˜m x dx dm m m = dm, k=m+1 m where 0 ≤ c˜m → 0, and thus 0 ≤ dm → 0. Now, from the last three relations, we obtain: 0 ≥ b/4 − dm.  Let us consider some consequences of the proposition. Corollary 1. Let α ∈ [0, 1/2] and 1/s = 1 + α. For a Banach space Y, suppose that there exist constants C > 0 such that for every natural n and for every n- dimensional subspace E of Y there exists a nite rank operator R in Y so that α ||R|| ≤ Cn and R|E = idE . Then Y ∈ APs. Corollary 2. Let α ∈ [0, 1/2] and 1/s = 1 + α. For a Banach space Y, suppose that there exist constants C > 0 such that for every natural n and for every n- dimensional subspace E of Y there exists a nite dimensional subspace F of Y, α containing E and Cn -complemented in Y. Then Y ∈ APs. Corollary 3. Let α ∈ [0, 1/2] and 1/s = 1 + α. For a Banach space Y, sup- pose that there exist constants C > 0 such that for every natural n and every α n-dimensional subspace E of Y is Cn -complemented in Y. Then Y ∈ APs. More- over, every subspace of the space Y has the APs.  GROTHENDIECK-LIDSKII THEOREM FOR SUBSPACES OF QUOTIENTS OF Lp-SPACES 4

It can be shown (but we do not need this in the note) that Y ∈ APs i for every ∗ b Banach space X the natural mapping X ⊗sY → L(X,Y ) is one-to-one (for other related results see, e.g., [11], [12]). Thus, taking this in account, we get: Corollary 4. In all above four assertions, in the case of Y with mentioned ∗ b properties, we have the quasi-Banach equality X ⊗sY = Ns(X,Y ), whichever the ∗ b space X was. In particular, Y ⊗sY = Ns(Y,Y ). Before giving more concrete applications of Proposition 1, let us mention the simplest example. Example 1. Let s ∈ (0, 1], p ∈ [1, ∞] and 1/s = 1 + |1/p − 1/2|. Any subspace as well as any factor space of any Lp-space have the property APs. We used this example in [13]. The statement of Example 1 follows from Corollary 4 and the results of D.R. Lewis (see [3], Corollary 4). As s matter of fact, one can get from the work [3] more general facts on com- plementability concerning Lp-situation. However, we prefer to consider abstract situations and to deal with spaces of nontrivial types and cotypes (partially, for using the results to be obtained in other considerations). We will apply mainly the results that can be found, e.g., in [1], [6], [8] and [9]. For the denitions of the notions of type and cotype, see any of this references (Rademacher type p = Gauss type p and Rademacher cotype q = Gauss cotype q; so, we can apply results from G. Pisier's lecture [9], assuming that we are working with Rademacher notions). Let us collect the facts we need. Proposition 2. Let X be a Banach space and 1 < p ≤ 2, 2 ≤ q < ∞. 1). If X is of type p (cotype p) then every subspace is of type p (cotype p); 2). [1, Proposition 11.11] If X is of type p then any quotient of X is of type p; 3). [1, Proposition 11.10] If X is of type p then X∗ is of cotype p′; 4). If X∗ is of type p then X is of cotype p′; 5). If X is of type p then any subspace of any quotient (and any quotient of any subspace) of X is of type p; 6). [1, Corollary 11.9] A Banach space has the same type or cotype as its bidual;

7). [1, Corollary 11.7] Each Lr-space (1 ≤ r < ∞) has type min{r, 2} and cotype max{r, 2}; 8). [9, see Theorem 4.1 and its Corollaries] There is a constant Dp,q > 0 such 1/p−1/q that every nite dimensional subspace E of X is Dp.q (dim E) -complemented in X. Recall also the well known general fact: in any Banach space every n-dimensional subspace is n1/2-complemented. We need in this note only the following immediate consequence of Proposition 2 and Corollary 3: Corollary 5. Let s ∈ (0, 1], p ∈ [1, ∞] and 1/s = 1 + |1/p − 1/2|. If a Banach space Y is isomorphic to a subspace of a quotient (or to a quotient of a subspace) of an Lp-space then it has the property APs. In particular, we get again (cf. [10] and see [2]):  GROTHENDIECK-LIDSKII THEOREM FOR SUBSPACES OF QUOTIENTS OF Lp-SPACES 5

Corollary 6 [2]. Every Banach space has the property AS2/3. Ÿ1. Main lemma. We are going to formulate to prove now the main lemma in this paper. It is interesting to note that in the proof we will use a part of the proof of Lemma from [13] as well as the statement of that Lemma itself. Let us recall the formulation of Lemma of [13]. Lemma 0. Let s ∈ (0, 1], p ∈ [1, ∞] and 1/s = 1 + |1/2 − 1/p|. Then the system of all eigenvalues (with their algebraic multiplicities) of any operator T ∈ Ns(Y,Y ), acting in any subspace Y of any Lp-space, belongs to the space l1. The same is true for the factor spaces of Lp-spaces. The next assertion contains this lemma 0 as a particular case. Lemma 1. Let s ∈ (0, 1], p ∈ [1, ∞] and 1/s = 1 + |1/2 − 1/p|. Then the system of all eigenvalues (with their algebraic multiplicities) of any operator T ∈ Ns(Y,Y ), acting in any subspace Y of any quotient of any Lp-space (equivalently: in any quotient Y of any subspace of any Lp-space), belongs to the space l1.

Proof of Lemma 1. Let p ∈ [1, ∞]. Let Y be a subspace of a quotient W (= Lp/V for some V ⊂ Lp) of an Lp-space and T ∈ Ns(Y,Y ) with an s-nuclear representation ∑∞ ′ ⊗ T = µkyk yk, ∑k=1 where || ′ || || || and ≥ ∞ s ∞ The operator can be factored yk , yk = 1 µk 0, k=1 µk < . T in the following way:

A ∆1−s j ∆s B T : Y −→ l∞ −→ lr ,→ c0 −→ l1 −→ Y, where and are linear bounded, is the natural injection, ∼ s and A B j ∆s (µk)k ∼ 1−s are the natural diagonal operators from into and from into ∆1−s (µk ) c0 l1 l∞ respectively. Here, is dened via the conditions | − | and ∑lr, r ∑ 1/s = 1 + 1/p 1/2 s ∞ we have to have (1−s)r ∞ for which − is good. k µk < : k µk < , (1 s)r = s Therefore, put 1/r = 1/s − 1, or 1/r = |1/p − 1/2|. Let Φ: Lp → W be a factor map, so that Y ⊂ W. Denote by Y0,Y0 ⊂ Lp, the preimage of under the map −1 Consider the operator | → Y Φ,Yo := Φ (Y ). Φ Y0 : Y0 Y (it is a factor map) and the following diagram:

| Φ Y0 A ∆1−s j ∆s B Y0 −→ Y −→ l∞ −→ lr ,→ c0 −→ l1 −→ Y. Since | is a factor map, we can nd a lifting → with | → Φ Y0 Q : l1 Y B = Φ Y0 Q : l1 Y0 → Y. Now, we get that the operator T can be factored as follows: | A ∆1−s j ∆s Q Φ Y0 T : Y −→ l∞ −→ lr ,→ c0 −→ l1 −→ Y0 −→ Y. Let → Then ∈ | ∈ U0 := Q∆sj∆1−sA : Y Y0. U0 Ns(Y,Y0),U := U0Φ Y0 Ns(Y0,Y0) and | ∈ By the principle of related operators(see [8], 6.4.3.2), T = Φ Y0 U Ns(Y,Y ). U and T have the same eigenvalues with the same algebraic multiplicities. But U acts  GROTHENDIECK-LIDSKII THEOREM FOR SUBSPACES OF QUOTIENTS OF Lp-SPACES 6 in a subspace Y0 of an Lp-space, so main Lemma of [13] can be applied. Therefore, by Lemma 0, Lemma 1 is proved. Corollary 7. If s ∈ (0, 1], p ∈ [1, ∞] with 1/s = 1 + |1/2 − 1/p| then the quasi-normed ideal Ψp,s is of (spectral) type l1. Ÿ1. Proof of Theorem We prefer to give here a complete proof although we could just refer to the proof of the corresponding theorem in [13] with giving some remarks.

Let Y be a subspace of a quotient of an Lp-space and T ∈ Ns(Y,Y ). By Corollary 5, we may (and do) identify the space Ns(Y,Y ) with the corresponding tensor prod- ∗ b ∗ b uct Y ⊗sY, which, in turn, is a subspace of the projective tensor product Y ⊗Y. Thus, the nuclear trace of T is well dened, and∑ we have to show that this trace of is just the spectral trace (= spectral sum) ∞ T n=1 λn(T ). By Lemma, the { }∞ of all eigenvalues of counting by mul- λn(T ) n=1 T, tiplicities, is in l1. Since the quasi-normed ideal Ψp,s is of spectral (= eigenvalue) type l1 (see Corollary 7), we can apply the main result from the paper [14] of M.C. White, which asserts:

(∗∗) If J is a quasi-Banach operator ideal with eigenvalue type l1, then the spec- tral sum is a trace on that ideal J. For the sake of completeness and to simplify the understanding, we (as in the paper [13]) give here some information about "trace" on an operator ideal. Namely, recall (see [8], 6.5.1.1, or Denition 2.1 in [14]) that a trace on an operator ideal J is a class of complex-valued functions, all of which one writes as τ, one for each component J(E,E), where E is a Banach space, so that (i) τ(e′ ⊗ e) = ⟨e′, e⟩ for all e′ ∈ E∗, e ∈ E; (ii) τ(AU) = τ(UA) for all Banach spaces F and operators U ∈ J(E,F ) and A ∈ L(F,E); (iii) τ(S + U) = τ(S) + τ(U) for all S, U ∈ J(E,E); (iv) τ(λU) = λτ(U) for all λ ∈ C and U ∈ J(E,E). Our operator T, evidently, belongs to the space Ψp,s(Y,Y ) and, as was said, Ψp,s is of eigenvalue type Thus, the assertion ∗∗ implies that the spectral sum ∑l1. ( ) λ, dened by ∞ for ∈ is a trace on λ(U) := n=1 λn(U) U Ψp,s(E,E), Ψp,s. By principle of uniform boundedness (see [7], 3.4.6 (page 152), or [5]), there exists a constant C > 0 with the property that

| | ≤ ||{ }|| ≤ λ(U) λn(U) l1 C νp,s(U)

for all Banach spaces E and operators U ∈ Ψp,s(E,E). Now, remembering that all operators in Ψp,s can be approximated by nite rank operators and taking in account the conditions (iii)(iv) for τ = λ, we obtain that the nuclear trace of our operator T coincides with λ(T ) (recall that the continuous trace is uniquely dened in such a situation; see [8], 6.5.1.2).  GROTHENDIECK-LIDSKII THEOREM FOR SUBSPACES OF QUOTIENTS OF Lp-SPACES 7 References

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[3] D.R. Lewis: Finite dimensional subspaces of Lp, Studia Math. 63 (1978), 207–212. [4] V. B. Lidski: Nonselfadjoint operators having a trace, Dokl. Akad. Nauk SSSR, 125(1959), 485487. [5] A. Pietsch: Eigenwertverteilungen von Operatoren in Banachrdumen, Hausdor-Festband: Theory of sets and topology, Berlin: Akademie-Verlag (1972), 391-402. [6] A. Pietsch: Operator Ideals, North Holland (1980). [7] A. Pietsch: Eigenvalues and s-numbers, Cambridge Univ. Press (1987). [8] A. Pietsch: History of Banach Spaces and Linear Operators, Birkhauser(2007). [9] G. Pisier: Estimations des distances aun espace euclidien et des constantes de projectiondes espaces de Banach de dimensions nie, Seminaire d'Analyse Fonctionelle 1978-1979, Centre de Math., Ecole Polytech., Paris (1979), expose10, 1-21. [10] O.I. Reinov: A simple proof of two theorems of A. Grothendieck, Vestn. Leningr. Univ. 7 (1983), 115-116. [11] O.I. Reinov: Disappearance of tensor elements in the scale of p-nuclear operators, Theory of operators and theory of functions (LGU) 1(1983), 145-165.

[12] O.I. Reinov: Approximation properties APs and p-nuclear operators (the case 0 < s ≤ 1), Journal of Mathematical Sciences 115, No. 3 (2003), 2243-2250.

[13] Oleg Reinov, Qaisar Latif: Grothendieck-Lidskii theorem for subspaces of Lp-spaces, Math. Nachr. 286, No. 2-3 (2013), 279 – 282. [14] M.C. White: Analytic multivalued functions and spectral trace, Math. Ann. 304 (1996), 665- 683.

Department of Mathematics and Mechanics, St. Petersburg State University, Saint Petersburg, RUSSIA. Abdus Salam School of Mathematical Sciences, 68-B, New Muslim Town, Lahore 54600, PAKISTAN. E-mail address: [email protected]

Abdus Salam School of Mathematical Sciences, 68-B, New Muslim Town, Lahore 54600, PAKISTAN. E-mail address: [email protected]